Aside from matching Google, which has offered a 64-bit version of Chrome since last August, Mozilla's native 64-bit build of Firefox boasts three key features: faster execution speed, better security, and the ability to run larger programs.
An advantage of using 64-bit applications is their use of a larger address space on a server or desktop. For a browser like Firefox, this translates into having more tabs open at once and running more ambitious in-browser applications. But other advantages of 64-bit apps include better leveraging of address space layout randomization (ASLR), a common technique for protecting against software exploits. Mozilla touts this as a boon for 64-bit Firefox, and Chrome's 64-bit edition has similar functionality.
Mozilla provided a big example for 64-bit asm.js: "browser-based games that deliver performant, native-like gameplay, such as those built with Epic Games' Unreal Engine," featuring assets that are "often much larger than we expect from traditional Web applications." With 64-bit address space at one's disposal, it's easier to load and process those assets, but there are ostensibly other applications for a larger address space beyond gaming.
Firefox was previously available in 64-bit editions for Mac OS X and Linux, but Windows still accounts for the majority of Firefox's market share. An earlier effort to put a 64-bit edition of Firefox into the hands of Windows users stalled in 2012, with the lack of 64-bit plug-ins for Firefox cited as a big reason.
But strong backlash from users prompted Mozilla to reconsider, although it has made no promises as to when a 64-bit version will be released to the general public. Native HTML5 functionality has eclipsed traditional plug-ins -- from replacing Flash to playing back encrypted video content -- so it's easier to deliver a native 64-bit browser.
Mozilla is holding off on delivering a 64-bit browser for users to focus on higher-priority projects -- for example, Firefox OS, set to debut next year on phones in major markets.